6.838: Shape Analysisgroups.csail.mit.edu/gdpgroup/assets/6838_spring_2021/1... · 2021. 2. 14. ·...
Transcript of 6.838: Shape Analysisgroups.csail.mit.edu/gdpgroup/assets/6838_spring_2021/1... · 2021. 2. 14. ·...
6.838:
Shape AnalysisJustin Solomon
Spring 2021
Instructor: Justin SolomonEmail: [email protected]
Geometric Data Processing Group:http://gdp.csail.mit.edu
Will cover administrative details over Zoom.
CodingJulia, Python, or Matlab preferred
MathFluency in linear algebra and multivariable calculus
Not required (won’t hurt):Graphics, differential geometry, numerics, ML
We want you to take this course!Assignments intended to be interesting
(may be unintentionally easy/hard!).
Image from postermywall.com
I. Theoretical toolbox
II. Computational toolbox
III. Application areas
I. Theoretical toolbox
II. Computational toolbox
III. Application areas
http://www.turbosquid.com/3d-models/medically-human-brain-3d-model/437885
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http://www.turbosquid.com/3d-models/medically-human-brain-3d-model/437885
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Spivak: A Comprehensive Introduction to Differential Geometry
http://en.wikipedia.org/wiki/Differential_geometry
Study of smooth manifolds
http://upload.wikimedia.org/wikipedia/commons/b/bc/Double_torus_illustration.png
http://www.sciencedirect.com/science/article/pii/S0010448510001983
Curvature and shape properties
Distances
Crane, Weischedel, Wardetzky. Geodesics in heat. TOG 2013.Wittman. Manifold learning techniques.
Flows and vector fields
Vaxman et al.Directional field synthesis, design, and processing.
EG STAR 2016.
Differential operators
Vallet and Lévy. Spectral Geometry Processing with Manifold Harmonics. EG 2008
“Enneahedra”
Only need angles and distanceshttp://upload.wikimedia.org/wikipedia/commons/2/2c/Hobo%E2%80%93Dyer_projection_SW.jpg
Same distance?
Only need angles and distanceshttp://www.phy.syr.edu/courses/modules/LIGHTCONE/pics/curved.jpg
Ant’s view
Heeren et al.Splines in the Space of Shells.
SGP 2016.
http://en.wikipedia.org/wiki/Double_pendulumhttp://www.ualberta.ca/dept/math/gauss/fcm/BscIdeas/SpcDmnsn/pndlm2.htm
Peyré, Cuturi, and Solomon.Gromov-Wasserstein Averaging of Kernel and Distance Matrices.
ICML 2016.
Tenenbaum et al.A Global Geometric Framework for Nonlinear Dimensionality Reduction.Science 2000.
https://www.researchgate.net/publication/274071948_Space_charge_and_resistive_wall_impedance_computation_in_the_frequency_domain_using_the_finite_element_methodhttp://blog.forty.to/wp-content/uploads/2013/05/Bars1.jpg
I. Theoretical toolbox
II. Computational toolbox
III. Application areas
Triangle mesh Triangle soup
Graph Point cloud
Pairwise distance matrix Dataset Network
Nearly anything with a notion of proximity/distance/curvature/…
• Collection of flat triangles
• Approximates a smooth surface
Can a triangle mesh have curvature?
Combine smooth and discrete
http://www.geomtop.org/teaching/conformal_2012Summer/title_image.jpg
Discretevs.
Discretized
Discrete theory parallelingdifferential geometry.
Structure preservation[struhk-cher pre-zur-vey-shuh n]:Keeping properties from the continuous abstraction exactly true in a discretization.
Images from: Grinspun and Secord, “The Geometry of Plane Curves” (SIGGRAPH 2006)
Convergence[kuh n-vur-juh ns]:Increasing approximation quality as a discretization is refined.
Can you have it all?
Fey et al. 30 Rock (2006-2013).
Pick and choosewhich properties you need.
But there is a huge toolbox of algorithms to draw from!
Chuang and Kazhdan. Fast Mean-Curvature Flow via Finite-Elements Tracking. CGF 2011.Coifman & Lafon. Diffusion Maps. ACHA 2006.
Smith and Schaefer. Bijective parameterization with free boundaries. SIGGRAPH 2015.
Bommes, Zimmer, Kobbelt. Mixed-integer quadrangulation. SIGGRAPH 2009.
Huang, Guibas. Consistent shape maps via semidefinite
programming. SGP 2013.
Krishnan, Fattal, Szeliski. Efficient preconditioning of
Laplacian matrices for computer graphics. SIGGRAPH 2013.
Palmer, Bommes, & Solomon.Algebraic Representations for Volumetric Frame Fields.TOG 2020.
I. Theoretical toolbox
II. Computational toolbox
III. Application areas
Graphics
Mertens et al. “Texture Transfer Using Geometry Correlation.”Fisher et al. “Context-Based Search for 3D Models.”
Mitra et al. “Symmetrization.”Bokeloh et al. “A connection between partial symmetry and inverse procedural modeling.”
Editing
Retrieval
Exploiting patterns
Transfer
Vision
http://eijournal.com/newsite/wp-content/uploads/2012/01/VELODYNE-IMAGE.jpgKim et al. “Multi-view image and tof sensor fusion for dense 3d reconstruction.”
Solomon et al. “Discovery of Intrinsic Primitives on Triangle Meshes.”Bronstein et al. “Three-Dimensional Face Recognition.”
Recognition
Segmentation
Navigation
Reconstruction
Medical Imaging
http://dmfr.birjournals.org/content/33/4/226/F3.large.jpghttp://www-sop.inria.fr/asclepios/software/inriaviz4d/SphericalImTransp.png
http://www.creatis.insa-lyon.fr/site/sites/default/files/segm2.png
Analysis
Registration
Segmentation
Manufacturing and Fabrication
http://www.conduitprojects.com/php/images/scan.jpghttp://www.emeraldinsight.com/content_images/fig/0330290204005.png
Scanning Defect detection
Architecture
Design and analysisVouga et al. “Design of self-supporting surfaces.”
Shape collectionsHuang et al. “Consistent shape maps via semidefinite programming.”
CorrespondenceOvsjanikov et al. “Functional maps.”
Deformation transferWeise et al. “Realtime performance-based facial animation.”
Simulation
Bergou et al. “Discrete viscous threads.”Wardetzky et al. “Discrete quadratic curvature energies.”
Scientific visualizationCrane et al. “Spin Transformations of Discrete Surfaces.”
Segmentation
http://www.bioinformaticslaboratory.nl/twiki/pub/EBioScience/News/freesurfer-3d.jpghttp://hal.inria.fr/docs/00/40/21/30/IMG/vivodtzev_et_al-Dagstuhl03.jpg
Machine learningZhu et al. Semi-Supervised Learning Using Gaussian Fields and Harmonic Functions. ICML 2003.
StatisticsHou et al. Novel semisupervised high-dimensional correspondences learning method. Opt. Eng. 2008.
6.838:
Shape AnalysisJustin Solomon
Spring 2021